A similarity transformation \({A\rightarrow BAB^{-1}}\) is equivalent to a change of the basis defining the vector components operated on by \({A}\), where the change of basis has matrix \({B^{-1}}\) so that \({v\rightarrow Bv}\)

The eigenvalues, determinant and trace of \({A}\) are independent of basis \({\Rightarrow}\) unchanged by a similarity transformation

The trace is a cyclic linear map: \({\textrm{tr}(ABC)=\textrm{tr}(BCA)=\textrm{tr}(CAB)}\)

The determinant is a multiplicative map: \({\textrm{det}(rAB)=r^{n}\textrm{det}(A)\textrm{det}(B)}\)

The trace equals the sum of eigenvalues; the determinant equals their product

Spectral theorem: a matrix is normal iff it can be diagonalized by a unitary similarity transformation; a real matrix is symmetric iff it can be diagonalized by an orthogonal similarity transformation

As previously noted, we can geometrically interpret an element of a matrix group with real entries as a transformation on \({\mathbb{R}^{n}}\). Such a transformation preserves the orientation of \({\mathbb{R}^{n}}\) if its determinant is positive, and preserves volumes if the determinant has absolute value one.

Any bilinear form \({\varphi}\) on \({\mathbb{R}^{n}}\) can be represented by a matrix in the standard basis, with the form operation then being \({\varphi(v,w)=v^{\textrm{T}}\varphi w}\). The group of matrices that preserve a form \({\varphi}\) consists of matrices \({A}\) that satisfy \({\varphi\left(Av,Aw\right)=\varphi\left(v,w\right)\Leftrightarrow\left(Av\right)^{\textrm{T}}\varphi(Aw)=v^{\textrm{T}}\varphi w\Leftrightarrow A^{\textrm{T}}\varphi A=\varphi}\). Any similarity transformation simply changes the basis of each \({A}\), leaving the group of matrices that preserve the form unchanged. Thus we can concern ourselves only with a canonical form of the preserved form. In \({\mathbb{R}^{n}}\), we have several naturally defined forms:

The Euclidean inner product, with canonical form \({I}\)

The pseudo-Euclidean inner product of signature \({(r,s)}\), with canonical form \({\left(r+s=n\right)}\)

Any matrix group defined as preserving one of these canonical forms then preserves all forms in the corresponding similarity class. Some matrix groups with entries in \({\mathbb{C}}\) or \({\mathbb{H}}\) can also be viewed as preserving a form in the vector space \({\mathbb{C}^{n}}\) or module \({\mathbb{H}^{n}}\), but we will mainly view these as linear transformations on \({\mathbb{R}^{2n}}\) or \({\mathbb{R}^{4n}}\).